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Has Zhang's work on bounded gaps between primes been extended to the following theorem?

For any arithmetic progression $an+b,\gcd(a,b)=1$, there is a constant $H$ (depending only on $a$) such that there are infinitely many prime pairs $p_1<p_2$ in this progression with $p_2-p_1<H$.

I have tried to look for this result online, the closest hit I got is this paper, which proves similar sounding statement.

How about extending this question to sets of primes of positive relative density (this extension and question as the whole inspired by answer to this question) and quantitative bounds?

Thanks in advance.

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Yes, see Deniz Ali Kaptan's recent arXiv preprint. (Added: See Terry Tao's comment below for more references.)

Concerning your second question, even full relative density is not enough to produce bounded gaps. Using standard upper bounds on the number of solutions $p'-p=d$ (for a given even $d>0$) it is surely straightforward to show that a certain full relative density subset of primes avoids bounded gaps (I have not checked the details though).

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    $\begingroup$ The stated result also appears in this paper of Benatar: arxiv.org/abs/1305.0348 , who also studies prime gaps in other sets of positive relative density. Other relevant literature includes the work of Thorner arxiv.org/abs/1401.6677 who studies Chebotarev sets (more general than arithmetic progressions) and Maynard arxiv.org/abs/1405.2593 (who studies dense clusters of primes in well-distributed subsets). Finally there is older work of Freiberg arxiv.org/abs/1110.6624 that uses the older GPY technology to obtain a related result. $\endgroup$
    – Terry Tao
    Aug 23, 2015 at 23:11
  • $\begingroup$ @TerryTao: Thank you for the more precise and more detailed information! $\endgroup$
    – GH from MO
    Aug 24, 2015 at 1:08
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    $\begingroup$ Thank you for this reply, and thank you @TerryTao for providing these references. $\endgroup$
    – Wojowu
    Aug 28, 2015 at 7:53

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